Basic annuity loan problem

Question

I've got a loan of $\$32000$, which I need to repay in $36$ monthly payments, with annual rate of $10\%$. What is the amount of monthly payment, which I need to pay?

There will be $36$ payments, thus $i=10\%/12=0.00833$. What's next step? I am new to this but I wish to understand it better. Thank you for help.

Answer 1

At the beginning you owe 32000. When do you make the first payment, immediately or after one month? Let's say after one month. If the payment amount is $a$, how much do you owe immediately after that first payment ... take into account the 32000 has increased by one month interest during the delay.

Once you can do that, apply the same method to find how much you owe after the second payment. (All these are in terms of $a$.) Continue this way, you get a finite series to add. The amount you owe after the last payment should be zero. That is an equation for the unknown value $a$.

But do not expect to solve that equation for $a$. There are tables (or nowadays computer programs) that will do it numerically for you.

Answer 2

Let: $$C:Capital=32000\\$$ $$I:Interest=10\\$$ $$ N:Number of months = 36\\$$ $$P:Payment\\$$ To find P, plug the numbers in the formula: $$P=\frac{C\times I\div 1200}{1-(1+(I\div1200))^{-N}}\\$$ And you'll find P = 1032.55